Problem: Evaluate the definite integral. $\int^{-\frac{5\pi}{2}}_{-2\pi}5\cos(x)\,dx = $
First, use the cosine rule: $\begin{aligned}\int^{-\frac{5\pi}{2}}_{-2\pi}5\cos(x)\,dx~&=~5\sin(x)\Bigg|^{-\frac{5\pi}{2}}_{{-2\pi}}\end{aligned}$ Second, plug in the limits of integration: $(5\cdot{\sin(-\frac{5\pi}{2})})-(5\cdot{\sin(-2\pi)}) = -5-0 =-5$. The answer: $\int^{-\frac{5\pi}{2}}_{-2\pi}5\cos(x)\,dx~=~-5$